Generalizations of elliptic Calogero-Moser systems — known as crystallographic elliptic Calogero-Moser systems — have been constructed for all crystallographic complex reflection groups [1]}. Our proposal is that these are candidates for Seiberg-Witten geometries. A nice feature of these integrable systems is that their full set of parameters matches the mass deformations of classes of \(4d\)  \(\mathcal {N}=2\)  quantum field theories.
For instance, we identify the elliptic Calogero-Moser systems attached to the crystallographic complex reflection groups of type \(G(m, 1, N)\)  with \(m=2, 3, 4, 6\)  as Seiberg-Witten integrable systems for \(4d\)  \(\mathcal {N}=2\)  rank \(N\)  \(D_4\)  and \(E_6\) , \(E_7\) , \(E_8\)  theories [2]}, [3]}, [4]}. Those theories belong to the the category of class-S theories, therefore their Seiberg-Witten integrable systems admit Hitchin system construction [5]}, [6]}, [7]}.
